Most advances in electronic spin-dependent non-adiabatic dynamics focus on refining the underlying dynamics methods. In contrast, this work considers an improved description of spin-orbit coupling by explicitly accounting for its relativistic origins. To this end, we extend a standard one-electron triatomic Jahn-Teller model to the four-component relativistic domain. In our formulation, the electron is treated using the Dirac-Coulomb Hamiltonian, while the nuclei remain non-relativistic, departing from the conventional formulation that incorporates the Pauli spin-orbit coupling as a perturbation. The most striking difference between our relativistic model and the conventional one is the presence of vibronic coupling terms on the anti-diagonal of the diabatic potential matrix. These terms scale as 1/c2 and, although nominally small, they can be amplified near avoided-crossings or in systems with heavy nuclei. Furthermore, their presence implies that nuclear motion can affect the direction of the electron spin, a feature entirely absent in the non-relativistic formulation of our model. In the adiabatic representation, the couplings translate to non-Abelian characteristics of the non-adiabatic coupling matrix that persist even in the Born-Oppenheimer approximation. Within our model setting, a relativistic formulation allows for a more intricate interplay between the electronic spin and nuclear degrees of freedom.